{"id":9252,"date":"2012-05-21T23:23:44","date_gmt":"2012-05-21T22:23:44","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9252"},"modified":"2021-12-29T15:16:24","modified_gmt":"2021-12-29T15:16:24","slug":"represente-na-forma-trigonometrica-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9252","title":{"rendered":"Represente na forma trigonom\u00e9trica"},"content":{"rendered":"<p><ul id='GTTabs_ul_9252' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9252' class='GTTabs_curr'><a  id=\"9252_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9252' ><a  id=\"9252_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_9252'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sendo $$\\begin{array}{*{20}{c}}<br \/>\n{z = \\sqrt 2\u00a0 &#8211; \\sqrt 2 i}&amp;{\\text{e}}&amp;{w =\u00a0 &#8211; \\frac{2}{3} + \\frac{2}{{\\sqrt 3 }}i}<br \/>\n\\end{array}$$ represente na forma trigonom\u00e9trica.<\/p>\n<ol>\n<li>$z$<\/li>\n<li>$w$<\/li>\n<li>$zw$<\/li>\n<li>$\\frac{z}{w}$<\/li>\n<li>${w^3}$<\/li>\n<li>$\\frac{1}{{ &#8211; w}}$<\/li>\n<li>${z^2}\\overline w $<\/li>\n<li>${z^4}:{w^3}$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9252' onClick='GTTabs_show(1,9252)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9252'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{z = \\sqrt 2\u00a0 &#8211; \\sqrt 2 i}&amp;{\\text{e}}&amp;{w =\u00a0 &#8211; \\frac{2}{3} + \\frac{2}{{\\sqrt 3 }}i}<br \/>\n\\end{array}$$<\/p>\n<\/blockquote>\n<ol>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\nz&amp; = &amp;{\\sqrt 2\u00a0 &#8211; \\sqrt 2 i} \\\\<br \/>\n{}&amp; = &amp;{2\\left( {\\frac{{\\sqrt 2 }}{{\\text{2}}} &#8211; \\frac{{\\sqrt 2 }}{2}i} \\right)} \\\\<br \/>\n{}&amp; = &amp;{2\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{4}} \\right)}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\nw&amp; = &amp;{ &#8211; \\frac{2}{3} + \\frac{2}{{\\sqrt 3 }}i} \\\\<br \/>\n{}&amp; = &amp;{\\frac{4}{3}\\left( { &#8211; \\frac{1}{{\\text{2}}} + \\frac{{\\sqrt 3 }}{2}i} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{4}{3}\\operatorname{cis} \\frac{{2\\pi }}{3}}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{zw}&amp; = &amp;{2\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{4}} \\right) \\times \\frac{4}{3}\\operatorname{cis} \\frac{{2\\pi }}{3}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{8}{3}\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{4} + \\frac{{2\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{8}{3}\\operatorname{cis} \\frac{{5\\pi }}{{12}}}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{zw}&amp; = &amp;{\\frac{{2\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{4}} \\right)}}{{\\frac{4}{3}\\operatorname{cis} \\frac{{2\\pi }}{3}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{2}{{\\frac{4}{3}}}\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{4} &#8211; \\frac{{2\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{3}{2}\\operatorname{cis} \\left( { &#8211; \\frac{{11\\pi }}{{12}}} \\right)}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{w^3}}&amp; = &amp;{{{\\left( {\\frac{4}{3}\\operatorname{cis} \\frac{{2\\pi }}{3}} \\right)}^3}} \\\\<br \/>\n{}&amp; = &amp;{{{\\left( {\\frac{4}{3}} \\right)}^3}\\operatorname{cis} \\left( {3 \\times \\frac{{2\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{64}}{{27}}\\operatorname{cis} \\left( 0 \\right)}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\frac{1}{{ &#8211; w}}}&amp; = &amp;{\\frac{1}{{ &#8211; \\frac{4}{3}\\operatorname{cis} \\frac{{2\\pi }}{3}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\operatorname{cis} \\left( 0 \\right)}}{{\\frac{4}{3}\\operatorname{cis} \\left( {\\frac{{2\\pi }}{3} + \\pi } \\right)}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{1}{{\\frac{4}{3}}}\\operatorname{cis} \\left( {0 &#8211; \\frac{{2\\pi }}{3} &#8211; \\pi } \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{3}{4}\\operatorname{cis} \\left( { &#8211; \\frac{{5\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{3}{4}\\operatorname{cis} \\frac{\\pi }{3}}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z^2}\\overline w }&amp; = &amp;{{{\\left( {2\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{4}} \\right)} \\right)}^2} \\times \\overline {\\frac{4}{3}\\operatorname{cis} \\frac{{2\\pi }}{3}} } \\\\<br \/>\n{}&amp; = &amp;{{2^2}\\operatorname{cis} \\left( { &#8211; 2 \\times \\frac{\\pi }{4}} \\right) \\times \\frac{4}{3}\\operatorname{cis} \\left( { &#8211; \\frac{{2\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{16}}{{\\frac{3}{3}}}\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{2} &#8211; \\frac{{2\\pi }}{3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{16}}{3}\\operatorname{cis} \\left( { &#8211; \\frac{{7\\pi }}{6}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{16}}{3}\\operatorname{cis} \\frac{{5\\pi }}{6}}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{z^4}:{w^3}}&amp; = &amp;{\\frac{{{{\\left( {2\\operatorname{cis} \\left( { &#8211; \\frac{\\pi }{4}} \\right)} \\right)}^4}}}{{{{\\left( {\\frac{4}{3}\\operatorname{cis} \\frac{{2\\pi }}{3}} \\right)}^3}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{16\\operatorname{cis} \\left( { &#8211; \\pi } \\right)}}{{\\frac{{64}}{{27}}\\operatorname{cis} 2\\pi }}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{16}}{{\\frac{{64}}{{27}}}}\\operatorname{cis} \\left( { &#8211; \\pi\u00a0 &#8211; 2\\pi } \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{27}}{4}\\operatorname{cis} \\left( { &#8211; 3\\pi } \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{27}}{4}\\operatorname{cis} \\pi }<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9252' onClick='GTTabs_show(0,9252)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sendo $$\\begin{array}{*{20}{c}} {z = \\sqrt 2\u00a0 &#8211; \\sqrt 2 i}&amp;{\\text{e}}&amp;{w =\u00a0 &#8211; \\frac{2}{3} + \\frac{2}{{\\sqrt 3 }}i} \\end{array}$$ represente na forma trigonom\u00e9trica. $z$ $w$ $zw$ $\\frac{z}{w}$ ${w^3}$ $\\frac{1}{{ &#8211; w}}$ ${z^2}\\overline&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19175,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,314,18],"series":[],"class_list":["post-9252","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-numeros-complexos-12--ano","tag-12-o-ano","tag-forma-trigonometrica","tag-numeros-complexos"],"views":1885,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat66.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9252","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=9252"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9252\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19175"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9252"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9252"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9252"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9252"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}